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PRYMVARIETIESOFSPECTRALCOVERS
TAMA´SHAUSELANDCHRISTIANPAULY
Abstract.
Givenapossiblyreducibleandnon-reducedspectralcover
π
:
X
→
C
overa
smoothprojectivecomplexcurve
C
wedeterminethegroupofconnectedcomponentsofthe
PrymvarietyPrym(
X/C
).Asanimmediateapplicationweshowthatthefinitegroupof
n
-
torsionpointsoftheJacobianof
C
actstriviallyonthecohomologyofthetwistedSL
n
-Higgs
modulispaceuptothedegreewhichispredictedbytopologicalmirrorsymmetry.Inparticular
thisyieldsanewproofofaresultofHarder–Narasimhan,showingthatthisfinitegroupacts
triviallyonthecohomologyofthetwistedSL
n
stablebundlemodulispace.
1.
Introduction
RecentlytherehasbeenrenewedinterestinthetopologyoftheHitchinfibration.TheHitchin
fibrationisanintegrablesystemassociatedtoacomplexreductivegroupGandasmooth
complexprojectivecurve
C
.ItwasintroducedbyHitchin[Hi]in1987,originatinginhisstudy
ofa2-dimensionalreductionoftheYang-Millsequations.In2006,KapustinandWitten[KW]
highlightedtheimportanceoftheHitchinfibrationfor
S
-dualityandtheGeometricLanglands
program.WhiletheworkofNgoˆ[N2]in2008showedthatthetopologyoftheHitchinfibration
isresponsibleforthefundamentallemmaintheLanglandsprogram.InNgoˆ’sworkandlater
intheworkofFrenkelandWitten[FW]acertainsymmetryoftheHitchinfibrationplaysan
importantrole.
InthispaperwefocusontheHitchinfibrationforthegroupG=SL
n
andforalinebundle
M
over
C
,i.e.,themorphism
nM(1)
h
:
M−→A
n
0
=
H
0
(
C,M
j
)
.
2=jHere
M
denotesthequasi-projectivemodulispaceofsemi-stableHiggsbundles(
E,φ
)over
C
ofrank
n
,fixeddeterminantΔandwithtrace-freeHiggsfield
φ
∈
H
0
(
C,
End
0
(
E
)
⊗
M
).Inthe
caseofSL
n
theabovementionedsymmetrygroupoftheHitchinfibrationisthePrymvarietyof
aspectralcover.Forthetopologicalapplicationsthedeterminationofitsgroupofcomponents
isthefirststep.Ngoˆworkswithintegral,thatisirreducibleandreduced,spectralcurves;but
itisinterestingtoextendhisresultstonon-integralcurves.Forreduciblebutreducedspectral
curvesitwasachievedbyChaudouardandLaumon[CL],whoprovedtheweightedfundamental
lemmabygeneralizingNgoˆ’sresultstoreducedspectralcurves.Inthispaperwedeterminethe
groupofconnectedcomponentsofthePrymvarietyfornon-reducedspectralcurvesaswell.
Inordertostatethemaintheoremweneedtointroducesom
S
enotation.Weassociatetoany
spectralcover
π
:
X
→
C
afinitegroup
K
asfollows:let
X
=
i
∈
I
X
i
beitsdecompositioninto
irreduciblecomponents
X
i
,let
X
ired
betheunderlyingreducedcurveof
X
i
,
m
i
themultiplicity
2000
MathematicsSubjectClassification.
Primary14K30,14H40,14H60.
1
2TAMA´SHAUSELANDCHRISTIANPAULY
of
X
ired
in
X
i
and
X
e
ired
thenormalizationof
X
i
.Wedenoteby
π
e
i
:
X
e
ired
→
C
theprojection
onto
C
andintroducethefinitesubgroups
K
i
=ker
π
e
i
∗
:Pic
0
(
C
)
−→
Pic
0
(
X
e
ired
)
⊂
Pic
0
(
C
)
,
aswellasthesubgroups(
K
i
)
m
i
=[
m
i
]
−
1
(
K
i
),where[
m
i
]denotesmultiplicationby
m
i
inthe
PicardvarietyPic
0
(
C
)parameterizingdegree0linebundlesover
C
.Finally,weput
\(2)
K
=(
K
i
)
m
i
⊂
Pic
0
(
C
)
.
I∈iWedenoteby
C
n
themultiplecurvewithtrivialnilpotentstructureoforder
n
havingunderlying
reducedcurve
C
.
WeconsiderthenormmapNm
X/C
:Pic
0
(
X
)
→
Pic
0
(
C
)betweentheconnectedcomponents
oftheidentityelementsofthePicardschemesofthecurves
X
and
C
anddefinethePrym
variety
Prym(
X/C
):=ker(Nm
X/C
)
.
Ourmainresultisthefollowing
Theorem1.1.
Let
π
:
X
→
C
beaspectralcoverofdegree
n
≥
2
.Withthenotationabovewe
havethefollowingresults:
(1)
Thegroupofconnectedcomponents
π
0
(Prym(
X/C
))
ofthePrymvariety
Prym(
X/C
)
equals
π
0
(Prym(
X/C
))=
K
b
,
where
K
b
=Hom(
K,
C
∗
)
isthegroupofcharactersof
K
.
(2)
Thenaturalhomomorphismfromthegroupof
n
-torsionlinebundles
Pic
0
(
C
)[
n
]
to
π
0
(Prym(
X/C
))
givenby
Φ:Pic
0
(
C
)[
n
]
−→
π
0
(Prym(
X/C
))
,γ
7→
[
π
∗
γ
]
,
where
[
π
∗
γ
]
denotestheclassof
π
∗
γ
∈
Pic
0
(
X
)
in
π
0
(Prym(
X/C
))
issurjective.In
particular,weobtainanupperboundfortheorder
|
π
0
(Prym(
X/C
))
|≤
n
2
g
,
where
g
isthegenusofthecurve
C
.
(3)
Themap
Φ
isanisomorphismifandonlyif
X
equalsthenon-reducedcurve
C
n
with
trivialnilpotentstructureoforder
n
.
Similardescriptionsof
π
0
(Prym(
X/C
))weregivenin[N1]inthecaseofintegralspectral
curvesandby[CL]inthecaseofreduciblebutreducedspectralcurves.Also[dCHM]use
specialcasesforSL
2
.
Foracharacteristic
a
∈A
n
0
wedenoteby
π
:
X
a
→
C
theassociatedspectralcoverofdegree
n
(seesection2.2)andby
K
a
thesubgroupofPic
0
(
C
)definedin(2)andcorrespondingtothe
cover
X
a
.LetΓ
⊂
Pic
0
(
C
)[
n
]beacyclicsubgroupoforder
d
ofthefinitegroupPic
0
(
C
)[
n
]of
n
-torsionlinebundlesover
C
andlet
A
0Γ
⊂A
n
0
denotetheendoscopicsublocusofcharacteristics
na
suchthattheassociateddegree
n
spectralcover
π
:
X
a
→
C
comesfromadegree
d
spectral
coveroverthee´taleGaloiscoverof
C
withGaloisgroupΓ(fortheprecisedefinitionseesection
5.1).Withthisnotationwehavethefollowing
Theorem1.2.
Wehaveanequivalence
Γ
⊂
K
a
⇐⇒
a
∈A
0Γ
.
3
Thisgivesadescriptionofthelocusofcharacteristics
a
∈A
n
0
suchthatthePrymvariety
00Prym(
X
a
/C
)isnon-connected,becauseclearly
A
Γ
2
⊂A
Γ
1
ifΓ
1
⊂
Γ
2
.
Corollary1.3.
Thesublocusofcharacteristics
a
∈A
n
0
suchthatthePrymvariety
Prym(
X
a
/C
)
isnotconnectedequalstheunion
[0(3)
A
endo
:=
A
Γ
,
where
Γ
variesoverallcyclicsubgroupsofprimeorderof
Pic
0
(
C
)[
n
]
.
Calculatingthedimensionsoftheendoscopicloci
A
0Γ
willleadtoanimmediatetopological
application.RecallthatPic
0
(
C
)[
n
]actson
M
bytensorization,andthiswillinduceanaction
ontherationalcohomology
H
∗
(
M
;
Q
).Wethenhave
Theorem1.4.
Let
n>
1
and
p
n
bethesmallestprimedivisorof
n
.Assumethat
M
=
K
C
,the
canonicalbundleof
C
,andthat
(
n,
deg(Δ))=1
.Thentheactionof
Pic
0
(
C
)[
n
]
on
H
k
(
M
;
Q
)
istrivial,providedthat
k
≤
2
n
2
(1
−
1
/p
n
)(
g
−
1)
.
Infactthisresultshouldbesharp,asthetopologicalmirrorsymmetryconjecture[HT,
Conjecture5.1]predictsthatthesmallestdegreewherePic
0
(
C
)[
n
]actsnon-triviallyis
k
=
n
2
(1
−
1
/p
n
)(2
g
−
2)+1
.
ThisresultshintsatthecloseconnectionbetweenNgoˆ’sstrategyin[N1,N2]forstudyingthe
symmetriesoftheHitchinfibrationandthetopologicalmirrorsymmetryconjecturesin[HT].
Morediscussiononthisconnectioncanbefoundin[Hau2].
Finallylet
N
denotethemodulispaceofstablevec